The Fast of the First Born is obviously traced back to the exodus. But like most things religious it is never that simple.
Some later Jewish groups tied this fast to a celebration of the salvation of the Jews from the plot of Haman. This is because Haman advanced his plot on the thirteenth of Nisan. It appears different Jewish groups instituted different requirements for the fast. In any case it is well established that Galatian Jews began the fast with the Last Supper.
What is called the last supper was on the night of the 14th, remember night comes before day to God, darkness then comes the Light, the truth. so if they ate it on the night of the 14th the 13th ended at sundown.
Which is a strange way to begin a fast, usually a fast if from sundown to sundown, they would have eaten after sundown on the 14th.